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Déterminant associé à une trace sur une algèbre de Banach

Pierre de La Harpe, Georges Skandalis (1984)

Annales de l'institut Fourier

Soient A une algèbre de Banach complexe, G L ∞ ( A ) le groupe général linéaire stable de A et G L ∞ 0 ( A ) sa composante connexe pour la topologie normique. Nous montrons que toute trace non nulle r : A → C permet de définir un homomorphisme Δ r de G L ∞ 0 ( A ) sur le quotient du groupe additif C par l’image r _ ( K 0 ( A ) ) du groupe de Grothendieck de A . Si A = M n ( C ) (respectivement si A est un facteur fini continu) avec la trace usuelle, alors exp ( i 2 π Δ r ) est le déterminant usuel (resp. exp ( Re ( i 2 π Δ r ) ) est celui de Fuglede et Kadison). Dans le cas général, les déterminants Δ r permettent...

Discontinuity of the product in multiplier algebras.

Mohamed Oudadess (1990)

Publicacions Matemàtiques

Entire functions operate in complete locally A-convex algebras but not continuously. Actually squaring is not always continuous. The counterexample we give is multiplier algebra.

Domination properties in ordered Banach algebras

H. du T. Mouton, S. Mouton (2002)

Studia Mathematica

We recall from [9] the definition and properties of an algebra cone C of a real or complex Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A. The Banach algebra A is then called an ordered Banach algebra. An important property that the algebra cone C may have is that of normality. If C is normal, then the order structure and the topology of A are reconciled in a certain way. Ordered Banach algebras have interesting spectral properties....

Dual Banach algebras: representations and injectivity

Matthew Daws (2007)

Studia Mathematica

We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C*- and W*-algebras; we show that interpolation space techniques can be used in place of GNS type arguments....

Dual complementors in topological algebras

Marina Haralampidou (2005)

Banach Center Publications

We deal with dual complementors on complemented topological (non-normed) algebras and give some characterizations of a dual pair of complementors for some classes of complemented topological algebras. The study of dual complementors shows their deep connection with dual algebras. In particular, we refer to Hausdorff annihilator locally C*-algebras and to proper Hausdorff orthocomplemented locally convex H*-algebras. These algebras admit, by their nature, the same type of dual pair of complementors....

Entire functions and equicontinuity of power maps in Baire algebras.

Abdellah El Kinani (2000)

Revista Matemática Complutense

We obtain that the power maps are equicontinuous at zero in any Baire locally convex algebra with a continuous product in which all entire functions operate; whence is m-convex in the commutative case. As a consequence, we get the same result of Mityagin, Rolewicz and Zelazko for commutative B0-algebras.

Equicontinuity of power maps in locally pseudo-convex algebras

Abdellah El Kinani (2003)

Commentationes Mathematicae Universitatis Carolinae

We show that, in any unitary (commutative or not) Baire locally pseudo-convex algebra with a continuous product, the power maps are equicontinuous at zero if all entire functions operate. We obtain the same conclusion if every element is bounded. An immediate consequence is a result of A. Arosio on commutative and complete metrizable locally convex algebras.

Eventually positive elements in ordered Banach algebras

Gerd Herzog, Peer C. Kunstmann (2023)

Commentationes Mathematicae Universitatis Carolinae

In ordered Banach algebras, we introduce eventually and asymptotically positive elements. We give conditions for the following spectral properties: the spectral radius belongs to the spectrum (Perron--Frobenius property); the spectral radius is the only element in the peripheral spectrum; there are positive (approximate) eigenvectors for the spectral radius. Recently such types of results have been shown for operators on Banach lattices. Our results can be viewed as a complement, since our structural...

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